It isnt fair to throw these two together, as the two places are so different climatically, but if you are going to travel down under you shouldn't miss either.
In Australia I found odd truffle-like fungi that took years to identify as Elaphomyces, a type of deer truffle that grows with eucalypts. Identified by Dr. James Trappe, who has studied the hypogeous fungi of Australia extensively.
Also, in New South Wales at a rest area I found these quite psychoactive psilocybes growing in duff. I assumed they were native.

In New Zealand, I found this unusual Cyttaria, a genus of Ascomycete unique to the antarctic beech ecology of parts of the southern hemisphere. The bright color attracts birds which eat them, and spores are dispersed with the bird's dung.

A common species of Armillaria that was present in wooded areas of Australia.

Wearoa, the tobacco pouch fungi, also found in the Andes of South America, is a stunning looking fungus. It is considered toxic, but I have no information on it.
This gigantic Cordyceps was found at the Kauri tree preserve, a rare patch of native New Zealand ecology. Many members of this group make interesting biochemicals.

This Gymnopilus, found in NSW, is not considered food.
